Consider an un-directed graph or network, \(\mathcal {G} =\lbrace V,E\rbrace \)  where \(V=\lbrace v_1,\ldots ,v_n\rbrace \)  is the set of vertices (nodes) and \(E = \lbrace (v_i, v_j) | v_i, v_j \in V \rbrace \)  is the set of edges (connections) which contains the unordered pairs of vertices. We denote the adjacency matrix corresponding to \(\mathcal {G}\)  as \({\bf A} \in \mathbb {R}^{n \times n}\)  which is defined as \(A_{ij} = 1\)  if nodes \(i\)  and \(j\)  are connected, and 0 otherwise. The \(|V|=N\)  and \(|E|=M\)  represent the number of nodes and number of edges in \(\mathcal {G}\) , respectively. The number of edges linked to a particular node \(v_i\)  is referred to as its degree and denoted by \(k_i = \sum _{j=1}^{n} a_{ij}\) . The average degree of the network is denoted by \(\langle k \rangle \)  = \(\frac{1}{n}\sum _{i=1}^{n} k_{i}\) .
We use two random graph models, the Erdős-Rényi (ER) and the Scale-Free (SF) networks, to model the coupled dynamical systems [1]}. The ER random model network or the graph-valued random variable with the parameters is denoted by \(\mathcal {G}^{ER}(N,p)\)  where \(N\)  is the number of nodes and \(p\)  is the edge probability [2]}. The existence of each edge is statistically independent of all other edges. When we refer to `the graph \(\mathcal {G}^{ER}(N, p)\) ', we mean one realization of the random variable with mean degree \(\langle k \rangle \)  and generated as follows. Starting with \(N\)  number of nodes, connecting them with a probability \(p=N/\langle k \rangle \) . The ER random network realization thus generated will have a Binomial degree distribution. The SF networks (\(\mathcal {G}^{SF}\) ) generated using the Barabási-Albert model follows a power-law degree distribution [1]}.
<FIGURE>